lecture04-dft - university of pennsylvania

2/7/18

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Lecture #4 – Converting from time to frequency domain

ESE 150 -- Spring 2018

!  Play this on piano:


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!  Play

4 quarter notes 1 whole note

Cheat: A5, B5, G5, G4, D5


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!  1s / quarter note ! 8s of sound !  How many bits to represent 8s of sound with

16b samples and 44KHz sampling? "  44K Hz x 16b/sample x 8s = 5632K =5Mbits


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!  How does musical staff represent sound? "  What does vertical position represent? "  Note shape?


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!  There are other ways to represent "  Frequency representation particularly efficient

880 988 784 392 587

Frequencies in Hertz


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!  How much information is this musical staff communicating?

!  How many keys on piano? " bits/note


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Hamburg Steinway D-274 Piano photo by Karl Kunde

https://commons.wikimedia.org/wiki/File:D274.jpg


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Larry Solomn: http://solomonsmusic.net/insrange.htm


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!  How much information is this musical staff communicating?

!  How many keys on piano? " bits/note !  Let’s say 8b duration !  How many bits for 5 notes?

"  (7b/note+8b/duration) x 5 note = 75 bits?


!  You reproduced 800 samples of a 300Hz sine wave at 1000Hz with 8b precision "  6400b

!  What did you need to specify to do that? !  How may bits to represent that?


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!  Can represent common sounds much more compactly in frequency domain than in time-sample domain "  Frequency domain ~ 75b "  Time-sample domain ~ 5Mb


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!  Teaser: frequency representation !  Where are we on course map? !  What we did in lab last week

"  How it relates to this week

!  The Fourier Series – can represent any signal !  The Discrete Fourier Transform (DFT) – can translate

"  Change of basis

!  Next Lab !  References

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NIC

10101001101

EULA ----------------click OK

NIC

CPU

A/D

sample

domain conversion

MP3 Player / iPhone / Droid

File- System

10101001101

compress freq pyscho-

acoustics

D/A

MIC

speaker

2 4

5,6

3

7,8,9

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Music

Numbers correspond to course weeks

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A/D

sample

domain conversion

MP3 Player / iPhone / Droid

compress freq

D/A

MIC

speaker

2 4 3

Music

Numbers correspond to course weeks

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10101001101

10101001101


Analog input

!  Week 1: Converted Sound to analog voltage signal !  a “pressure wave” that changes air molecules w/ respect to time !  a “voltage wave” that changes amplitude w/ respect to time

!  Week 2: Sampled voltage, then quantized it to digital sig. "  Sample: Break up independent variable, take discrete ‘samples’ "  Quantize: Break up dependent variable into n-levels (need 2n bits to digitize)

!  Week 3: Compress digital signal "  Use even less bits without using sound quality!

!  Week 4 (upcoming): Before we compress… "  Put our ‘digital’ data into another form…BEFORE we compress…less stuff to compress!

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ADC Digital Output

compress


Background

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!  With this compact notation "  Could communicate a sound to pianist "  Much more compact than 44KHz time-sample

amplitudes (fewer bits to represent) "  Represent frequencies

ESE 150 -- Spring 2018 18

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! 

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Time domain representation Frequency domain representation


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Above is called an “amplitude spectrum” plot, we also have “phase spectrum” plots in the frequency domain


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ESE 150 -- Spring 2018 21

!  Another example

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The time domain plot on the right is really the sum of 5 sinusoids, where 5 Hz is the strongest component of the signal

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ESE 150 -- Spring 2018 23

The frequency domain &

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!  Fourier series: "  Any periodic signal can

be represented as a sum of simple periodic functions: sin and cos

sin(nt) and cos(nt) where n = 1, 2, 3, …

These are called the harmonics of the signal

ESE 150 -- Spring 2018 25

Jean Baptiste Joseph Fourier, wikipedia

!  States that any PERIODIC function can be written as a sum of sines and cosines! "  This is huge! "  It also gives us a way to breakup a periodic function into its sine/

cosine parts

!  Why do we care? "  Well if music was periodic, we could break it into its sine/cosine

pieces and store only the amp/freq/phase #  Huge storage advantage

"  But also, once we break it down, we could modify them #  Say we want to enhance the bass or treble of a signal? #  By increasing amplitude of bass frequencies, we’ll get more bass

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!  Other implications… "  If a signal is periodic, we can convert it from time to frequency

!  Imagine anything periodic "  If Earthquake vibrations could be separated into sinusoids…

#  Think vibrations at different speeds and strengths "  We could design buildings/bridges to avoid interacting with the strongest

frequency vibrations (and not just “all of them”)

"  If a radio-wave is our signal and its periodic #  We could hone in on a particular “channel” and listen only to that one…like a radio/TV!

"  It has endless applications in: communications, astronomy, geology, optics

!  There is one problem… "  The above examples aren’t exactly periodic! "  But we’ll address that soon…first let’s see the Fourier series

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The Fourier Theorem states that any periodic function f(t) of period T can be cast in the form:


€

f (t) = ao + an cos(nω ot) + bn sin(nω ot)( )n=1

∞

∑

•  The cos(nx) and sin(nx) functions form an orthogonal basis: •  allow us to represent any periodic

signal by taking a linear combination •  of the basis functions without

interfering with one another •  AKA: superposition works!

ESE 150 -- Spring 2018 29

!  What is an orthogonal basis? !  Example: 3D space

"  Basis: [i j k] "  Linear combination: xi + yj + zk "  Coordinate representation: [x y z]

!  Example: Quadratic polynomials "  Basis: [x2 x 1] "  Linear combination: ax2 + bx + c "  Coordinate representation: [a b c]

!  All vectors in the space can be represented as sequences of coordinates, or ordered coefficients of the basis vectors "  This is what the Fourier Series does…

#  just uses cos(nt) and sin(nt) as its “Basis”

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"  Consider the xy-plane:

"  We can address a point by its two coordinates (x,y) #  the grey point is located at (10, 3)

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0 5 10 0

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•  In Fourier series, we have a basis with infinite dimension! •  so we sum up over an infinite number of harmonics •  harmonics are 1, cos(t), cos(2t), cos(3t), …, sin(t), sin(2t),

…


! 


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n→∞

ESE 150 -- Spring 2018 33

(falstad.com/fourier/) ESE 150 -- Spring 2018

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(falstad.com/fourier/)

!  The idea of the series: "  Any PERIODIC wave can be represented as simple sum

of sine waves !  2 Caveats:

"  Linearity: # The series only holds while the system it is describing is linear

because it relies on the superposition principle #  -aka – adding up all the sine waves is superposition in action

"  Periodicity: # The series only holds if the waves it is describing are periodic # Non-periodic waves are dealt with by the Fourier Transform

$  We will examine that in the 2nd half of lecture

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!  Remember we said we needed to sample at twice the maximum frequency "  Now see all signals can be represented as a linear sum

of frequencies

"  …and the frequency components are orthogonal # Can be extracted and treated independently


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!  Close Encounters Mothership !  https://www.youtube.com/watch?

v=S4PYI6TzqYk


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!  In the first half of the lecture we introduced: "  The idea of frequency domain "  The Fourier Series

!  In the second half of the lecture: "  Fourier Transform "  See how to perform this time-frequency translation "  Examples

ESE 150 -- Spring 2018 39

!  Compute Dot Products with frequencies "  (equivalently, compute Fourier Components)


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i 0 1 2 3 4 5 6 7 8 9 10 time 0.0 0.1 0.2 0.3 0.4

0.5

0.6 0.7 0.8 0.9 1.0

Sample 0 0.95 0.59 -0.59 -0.95 0 0.95 0.59 -0.59 -0.95 0

i 0 1 2 3 4 5 6 7 8 9 10 k=1 0.00 0.59 0.95 0.95 0.59 0.00 -0.59 -0.95 -0.95 -0.59 0.00 k=2 0.00 0.95 0.59 -0.59 -0.95 0.00 0.95 0.59 -0.59 -0.95 0.00 k=3 0.00 0.95 -0.59 -0.59 0.95 0.00 -0.95 0.59 0.59 -0.95 0.00 k=4 0.00 0.59 -0.95 0.95 -0.59 0.00 0.59 -0.95 0.95 -0.59 0.00 k=5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

€

2N

Sample[i] × sin 2πkiN

⎛

⎝ ⎜

⎞

⎠ ⎟

i=0

N∑

!  Can identify frequencies with dot product "  Identifying projection onto each basis vector

in Fourier Series

!  Works because they are orthogonal

!  Performing a change of basis "  From time-sample basis "  To Fourier (sine, cosine) basis


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Another choice of bases. Change of basis.

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Euler’s Identity: Fourier Series:

Euler’s Identity in the Fourier Series:


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f (t) = ao + an cos(nω ot) + bn sin(nω ot)( )n=1

∞

∑

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Fourier Series

Fourier Transform – for continuous functions: Input: Time Domain Output: Frequency Domain


!  Instead of thinking only in sinusoids… "  Complex sinusoids can be thought of as cycles… "  Let’s visual complex #s a bit… "  http://betterexplained.com/examples/fourier/

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!  How to make a song appear “periodic:” "  Treat the entire song as 1 period of a very complicated

sinusoid! "  This is the assumption of the Fourier Transform

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Fourier Series – decomposes f(t) into a “weighted sum” of sine waves We only need to store amplitude and frequency of each harmonic of the fund. freq.

Fourier Series says it can be broken into simple sine waves

Fourier Series for this f(t):

Fourier Series in general form:


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Periodic Function in Time Domain

Fourier Series for this f(t):

Alternate view of Fourier Series …converting signals in the “time domain” to the “frequency domain”

Periodic Function in Frequency Domain

1st Fundamental amplitude: 0.5 2nd Fundamental amplitude: 1 And so on…

Store only frequency information:


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! 

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Discrete Time to Frequency:

Frequency back to Time:


!  A DFT transforms N samples of a signal in time domain "  into a (periodic) frequency representation with N samples "  So we don’t have to deal with real signals anymore

!  We work with sampled signals (quantized in time), "  and the frequency representation we get is also quantized in time!


50 (sepwww.stanford.edu/oldsep/hale/FftLab.html)

Music Signal in Discrete Time !

Music Signal “Transformed” To Frequency Domain !

!  A smaller sampling period means: -> more points to represent the signal larger N -> more harmonics used in DFT N harmonics -> Smaller error compared to actual analog

signal we capture/produce !  DFTs are extensively used in practice, since

computers can handle them ESE 150 -- Spring 2018

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!  A signal sampled in time can be approximated arbitrarily closely from the time-sampled values

!  With a DFT, each sample gives us knowledge of one harmonic

!  Each harmonic is a component used in the reconstruction of the signal

!  The more harmonics we use, the better the reconstruction

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{ Cos[0t], Sin[1t], Cos[1t] , Sin[2t], Cos[2t] , Sin[3t], Cos[3t] }


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7 Samples; 7 Harmonics 11 Samples 15 Samples; 15 Harmonics ; 11 Harmonics

DFT DFT DFT

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t

v

Measured Data:

Sampled Signal:


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!  Approximate Reconstruction "  although always achievable "  may require a lot of samples "  to get good performance

Sometimes time is better than frequency!

ESE 150 -- Spring 2018 55

15 Samples & Harmonics



!  However, N can get very large "  e.g., with a sampling rate of 48,000Hz "  How big is N for a 4 minute song?

!  How many operations does this translate to? "  To compute one frequency component? "  To compute all N frequency components?

!  This is not practical. Instead, we use a window of values to which we apply the transform "  Typical size: …, 512, 1024, 2048, …

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!  In the example below, we traverse the signal but only look at 64 samples at a time

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Time

Freq

(sepwww.stanford.edu/oldsep/hale/FftLab.html)

!  Not really connect-the-dots in time !  Recall near Nyquist rate

"  Could often miss the peak "  Get poor sine waves

# …look like peak moves around even if sampled above Nyquist rate

!  Better reconstruction "  Convert to frequency

# Which can perfectly represent up to half sampling rate

"  Reconstruct from frequency basis


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!  Can represent signals in frequency domain "  Different basis – basis vectors of sines and cosines

!  Often more convenient and efficient than time domain "  Remember musical staff

!  Can convert between time and frequency domain "  Using a dot-product to calculate time or frequency

components


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!  Lab 4: "  You will identify Frequency components using FFT in

Matlab

!  Remember: "  Lab 3 report is due on Friday

ESE 150 -- Spring 2018 60

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!  ESE325 – whole course on Fourier Analysis !  ESE224 – signal processing !  ESE215, 319, 419 – reason about behavior of

circuits in time and frequency domains


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!  S. Smith, “The Scientists and Engineer’s Guide to Digital Signal Processing,” 1997.

!  https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

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